11 REGULARIZATION 303 (b) Show that if the analytic function FM is identically 0 on the half-plane Sz> 0, then p, = 0. (Replacing ft by a regularization p, * (/• /3), show that FJ.*./ f — z and use (a) to conclude that JJL * (/• f$) is zero.) 17. Let G be a holomorphic function in the disk B: | z < 1. Show that ^G(z) ^ 0 for all z e B if and only if there exists a positive measure v on the interval [0, 2?r] such that J2it gi<P _ o ^ where c e R. (To show that the condition is necessary, note that for \z\ < r < 1 we have where pr(q>) = 0$G(rei(f>). Observe that the measure^ • A (where A is Lebesgue measure on [0, 2?r]) is positive and of total mass ^G(O), and use (13.4.3).) Deduce that for a function F holomorphic in the upper half-plane Sz > 0 to be such that ^F(z) 2> 0 for all z in this half-plane, it is necessary and sufficient that there should exist a bounded positive measure /x on R and two constants a ^ 0, b E R such that — z for ,/z > 0. (Map B onto the upper half-plane by means of znW(l + z)/(l — z)). 18. (a) Let A be Lebesgue measure on R, and let / be a A-integrable function. For each h > 0, put fJx\ —__ I f(x — t^dt Jh\-X-J — . J \X l)Ul. Show that, for almost all x e R, fh(x)-^-f(x) as /z~>0 + (Lebesgue's theorem). (If we put (Section 12.7, Problem 8) then R(/-^) = R(/) for all g £ Jf(R); also R(/) ^ 5(/) + |/|, with the notation of Section 14.10, Problem 5. Use Section 14,10, Problem 5(c) to deduce that the set of x e R such that R(/)(*) > a has measure zero, for all a > 0.) (b) Let (ffn) be a sequence of nonnegative functions in «#"(R), satisfying conditions (a), (b), and (c) of (14.11.1); assume also that#rt(-0 = #«(0 and that#n is decreasing in [0, •+• oo [. Show that as n ~* oo, (gn * /)(x) ->/W almost everywhere in R (Lebesgue's theorem). (Same method, using Section 14.10, Problem 5(d).) (c) State and prove analogous results for Haar measure on the torus T.ondition is sufficient, consider a sequence (un) of functions belonging to «#" +(G)